Mathematics for the interested outsider

Okay, a couple weeks ago I defined a chain complex to be a sequence with the property that . The maps are called the “differentials” of the sequence. As usual, these are the objects of a category, and we now need to define the morphisms.

Consider chain complexes and . We will write the differentials on as and those on as . A chain map is a collection of arrows that commute with the differentials. That is, . That these form the morphisms of an -category should be clear.

Given two chain complexes with zero differentials — like those arising as homologies — any collection of maps will constitute a chain map. These trivial complexes form a full -subcategory of the category of all chain complexes.

We already know how the operation of “taking homology” acts on a chain complex. It turns out to have a nice action on chain maps as well. Let’s write for the kernel of and for the image of , and similarly for . Now if we take a member (in the sense of our diagram chasing rules) so that , then clearly . That is, if we restrict to , it factors through . Similarly, if there is a with , then , and thus the restriction of to factors through .

So we can restrict to get an arrow which sends the whole subobject into the subobject . Thus we can pass to the homology objects to get arrows . That is, we have a chain map from to . Further, it’s straightforward to show that this construction is -functorial — it preserves addition and composition of chain maps, along with zero maps and identity maps.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.